Rewrite the irrational cube root 189−−−√3 as the product of an integer and another irrational cube root. Show your work. Note: You will find the option to add a cubic root symbol in the Algebra ( × ) keyboard.

Answer 1

To rewrite the irrational cube root \(\sqrt[3]{189}\) as the product of an integer and another irrational cube root, we will start by factoring 189 into its prime components:

1. Factor 189:

   • Start with \(189\) and check for divisibility by \(3\): \[ 189 \div 3 = 63 \]
   • \(63\) can again be divided by \(3\): \[ 63 \div 3 = 21 \]
   • Finally, \(21\) can also be divided by \(3\): \[ 21 \div 3 = 7 \]

   So, we have the complete factorization: \[ 189 = 3^3 \times 7 \]

2. Apply cube root: Now we apply the cube root: \[ \sqrt[3]{189} = \sqrt[3]{3^3 \times 7} \]

3. Separate into products: We can use the property of cube roots, which states \(\sqrt[3]{a \times b} = \sqrt[3]{a} \times \sqrt[3]{b}\): \[ \sqrt[3]{189} = \sqrt[3]{3^3} \times \sqrt[3]{7} \]

   Calculating \(\sqrt[3]{3^3}\) gives us \(3\): \[ \sqrt[3]{189} = 3 \times \sqrt[3]{7} \]

Thus, we can rewrite \(\sqrt[3]{189}\) as the product of an integer and another cube root: \[ \sqrt[3]{189} = 3 \times \sqrt[3]{7} \]

Final result: \[ \sqrt[3]{189} = 3 \sqrt[3]{7} \]

Answer 2

can you put it in plain language

Answer 3

Sure! Here's a step-by-step explanation in plain language:

1. Factor 189: Break down the number 189 into smaller numbers that multiply together to give 189. First, we can divide 189 by 3, which gives us 63. We can keep dividing by 3:

   • 63 divided by 3 is 21.
   • 21 divided by 3 is 7.

   So, the complete breakdown is: \(189 = 3 \times 3 \times 3 \times 7\) or \(189 = 3^3 \times 7\).

2. Take the cube root: When we want to find the cube root of 189 (which means what number multiplied by itself three times gives 189), we can use the fact that \(\sqrt[3]{a \times b} = \sqrt[3]{a} \times \sqrt[3]{b}\).

   So we can write: \[ \sqrt[3]{189} = \sqrt[3]{3^3 \times 7}. \]

3. Simplify the cube root: The cube root of \(3^3\) is simply \(3\) because \(3 \times 3 \times 3 = 27\) (which is what we want).

   Putting it all together, we get: \[ \sqrt[3]{189} = 3 \times \sqrt[3]{7}. \]

Conclusion: We have rewritten the cube root of 189 as \(3\) multiplied by the cube root of \(7\). So, the answer is: \[ \sqrt[3]{189} = 3 \sqrt[3]{7}. \]